On a class of degenerate elliptic equations in weighted Hölder spaces

Abstract

We study the Dirichlet problem for the degenerate elliptic equations \[ P_0\Delta\,u+\gamma(\nabla\,P_0,\nabla\,u)=f(x,u,\nabla\,u)\quad \mbox{in $\Omega$},\qquad \mbox{$u=0$ on $\partial \Omega$}, \] where $\gamma\geq 0$ is a given parameter, $\Omega\subset \mathbb{R}^n$ is an annular region, the given function $P_0(x)$ is such that $|\nabla\,P_0|+P_0\geq \epsilon> 0$ in $\overline\Omega$, and $P_0=0$ on the outer boundary of $\Omega$. The equation is degenerate elliptic when $\gamma>0$, while for $\gamma=0$ it transforms into the classical Poisson equation. We introduce the weighted Hölder spaces suitable for the study of the problem throughout the range of the parameter $\gamma\geq 0$. We derive the Schauder-type estimates and prove the existence of a unique classical solution. It is shown that in the case $\gamma>0$ the solution of the degenerate equation and the given function $P_0$ possess the same regularity properties. In the case $\gamma=0$ (the Poisson equation) the regularity of $u$ is better than the regularity of $P_0$. The proof is based on a new method of estimating the derivatives of solutions of the Poisson equation near the boundary of the problem domain which requires neither differentiation of the equation, nor straightening the boundary.